Fast parallel matrix and GCD computations
01 Apr 1982-Vol. 52, Iss: 3, pp 65-71
TL;DR: Two parallel algorithms to compute the determinant and characteristic polynomial of n×n-matrices and the gcd of polynomials of degree ≤n and a fast parallel Las Vegas algorithm for the rank of matrices are presented.
Abstract: We present parallel algorithms to compute the determinant and characteristic polynomial of n×n-matrices and the gcd of polynomials of degree ≤n. The algorithms use parallel time O(log2n) and a polynomial number of processors. We also give a fast parallel Las Vegas algorithm for the rank of matrices. All algorithms work over arbitrary fields.
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TL;DR: An attempt is made to identify important subclasses of NC and give interesting examples in each subclass, and a new problem complete for deterministic polynomial time is given, namely, finding the lexicographically first maximal clique in a graph.
Abstract: The class NC consists of problems solvable very fast (in time polynomial in log n ) in parallel with a feasible (polynomial) number of processors. Many natural problems in NC are known; in this paper an attempt is made to identify important subclasses of NC and give interesting examples in each subclass. The notion of NC 1 -reducibility is introduced and used throughout (problem R is NC 1 -reducible to problem S if R can be solved with uniform log-depth circuits using oracles for S ). Problems complete with respect to this reducibility are given for many of the subclasses of NC . A general technique, the “parallel greedy algorithm,” is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in NC 2 (solvable by uniform Boolean circuits of depth O (log 2 n ) and polynomial size). The class LOGCFL is given a new characterization in terms of circuit families. The class DET of problems reducible to integer determinants is defined and many examples given. A new problem complete for deterministic polynomial time is given, namely, finding the lexicographically first maximal clique in a graph. This paper is a revised version of S. A. Cook, (1983, in “Proceedings 1983 Intl. Found. Comut. Sci. Conf.,” Lecture Notes in Computer Science Vol. 158, pp. 78–93, Springer-Verlag, Berlin/New York).
686 citations
01 Jan 1987
TL;DR: A new algorithm for finding a maximum matching in a general graph that its only computationally non-trivial step is the inversion of a single integer matrix, the isolating lemma, and other applications to parallel computation and randomized reductions are shown.
Abstract: We present a new algorithm for finding a maximum matching in a general graph. The special feature of our algorithm is that its only computationally non-trivial step is the inversion of a single integer matrix. Since this step can be parallelized, we get a simple parallel (RNC 2) algorithm. At the heart of our algorithm lies a probabilistic lemma, the isolating lemma. We show other applications of this lemma to parallel computation and randomized reductions.
667 citations
Book•
06 Apr 1995
TL;DR: In providing an up-to-date survey of parallel computing research from 1994, Topics in Parallel Computing will prove invaluable to researchers and professionals with an interest in the super computers of the future.
Abstract: This volume provides an ideal introduction to key topics in parallel computing. With its cogent overview of the essentials of the subject as well as lists of P -complete- and open problems, extensive remarks corresponding to each problem, a thorough index, and extensive references, the book will prove invaluable to programmers stuck on problems that are particularly difficult to parallelize. In providing an up-to-date survey of parallel computing research from 1994, Topics in Parallel Computing will prove invaluable to researchers and professionals with an interest in the super computers of the future.
533 citations
Book•
09 Dec 2010TL;DR: The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what it finds to be the most interesting and accessible research directions, with an emphasis on works from the last two decades.
Abstract: A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions.
The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we discuss the recent lower bounds for multilinear circuits and formulas, the advances in the question of deterministically checking polynomial identities, and the results regarding reconstruction of arithmetic circuits. We do, however, also cover part of the classical works on arithmetic circuits. In order to keep this monograph at a reasonable length, we do not give full proofs of most theorems, but rather try to convey the main ideas behind each proof and demonstrate it, where possible, by proving some special cases.
509 citations
TL;DR: A new algorithm for finding a maximum matching in a general graph with special feature is that its only computationally non-trivial step is the inversion of a single integer matrix, the isolating lemma.
Abstract: We present a new algorithm for finding a maximum matching in a general graph. The special feature of our algorithm is that its only computationally non-trivial step is the inversion of a single integer matrix. Since this step can be parallelized, we get a simple parallel (RNC
2) algorithm. At the heart of our algorithm lies a probabilistic lemma, the isolating lemma. We show other applications of this lemma to parallel computation and randomized reductions.
506 citations
References
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01 Jan 1959
TL;DR: The theory of normal subgroups and homomorphisms was introduced in this article, along with the theory of $p$-groups regular $p-groups and their relation to abelian groups.
Abstract: Introduction Normal subgroups and homomorphisms Elementary theory of abelian groups Sylow theorems Permutation groups Automorphisms Free groups Lattices and composition series A theorem of Frobenius solvable groups Supersolvable and nilpotent groups Basic commutators The theory of $p$-groups regular $p$-groups Further theory of abelian groups Monomial representations and the transfer Group extensions and cohomology of groups Group representation Free and amalgamated products The Burnside problem Lattices of subgroups Group theory and projective planes Bibliography Index Index of special symbols.
2,960 citations
2,331 citations
01 May 1978
TL;DR: A model of computation based on random access machines operating in parallel and sharing a common memory is presented and can accept in polynomial time exactly the sets accepted by nondeterministic exponential time bounded Turing machines.
Abstract: A model of computation based on random access machines operating in parallel and sharing a common memory is presented. The computational power of this model is related to that of traditional models. In particular, deterministic parallel RAM's can accept in polynomial time exactly the sets accepted by polynomial tape bounded Turing machines; nondeterministic RAM's can accept in polynomial time exactly the sets accepted by nondeterministic exponential time bounded Turing machines. Similar results hold for other classes. The effect of limiting the size of the common memory is also considered.
951 citations
TL;DR: In this paper, the authors present a deterministic procedure for factoring polynomials over finite fields, which reduces the problem of factoring an arbitrary polynomial over the Galois field GF(p m) to finding the roots in GF(m) of certain other polynomorphisms over GF (m).
Abstract: This paper reviews some of the known algorithms for factoring polynomials over finite fields and presents a new deterministic procedure for reducing the problem of factoring an arbitrary polynomial over the Galois field GF(p m) to the problem of finding the roots in GF(p) of certain other polynomials over GF(p). The amount of computation and the storage space required by these algorithms are algebraic in both the degree of the polynomial to be factored and the logarithm of the order of the finite field. Certain observations on the application of these methods to the factorization of polynomials over the rational integers are also included.
506 citations